[tex]\bf \sqrt{\cfrac{50x^6y^3}{9x^8}}=\cfrac{5y^c\sqrt{2y}}{dx}\\\\
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\sqrt{\cfrac{50x^6y^3}{9x^8}}~~
\begin{cases}
50=2\cdot 25\\
\qquad 2\cdot 5^2\\
x^6=x^{3\cdot 2}\\
\qquad (x^3)^2\\
y^3=y^{2+1}\\
\qquad y^2\cdot y\\
9=3^2\\
x^8=x^{4\cdot 2}\\
\qquad (x^4)^2
\end{cases}\implies \cfrac{\sqrt{50x^6y^3}}{\sqrt{9x^8}}\implies \cfrac{\sqrt{2\cdot 5^2(x^3)^2\cdot y^2\cdot y}}{\sqrt{3^2(x^4)^2}}[/tex]
[tex]\bf \cfrac{5x^3y\sqrt{2y}}{3x^4}\implies \cfrac{5y\sqrt{2y}}{3x^4x^{-3}}\implies \cfrac{5y\sqrt{2y}}{3x^{4-3}}\implies \cfrac{5y^{\stackrel{\downarrow }{1}}\sqrt{2y}}{\stackrel{\downarrow }{3} x}[/tex]
